Spherically symmetric configurations in the quadratic f (R) gravity

We study spherically symmetric configurations of the quadratic f (R) gravity in the Einstein frame. We found the global qualitative behavior of the metric and the scalaron field for all static solutions satisfying the conditions of asymptotic flatness. These solutions are proved to be regular everywhere except for a naked singularity at the center; they are uniquely determined by the total mass M and the ``scalar charge'' Q characterizing the strength of the scalaron field at spatial infinity. The case Q=0 yields the Schwarzschild solution, but an arbitrarily small Q 0 leads to the appearance of a central naked singularity having a significant effect on the neighboring region, even though the space-time metric in the outer region is practically insensitive to the scalaron field. Approximation procedures are developed to derive asymptotic relations near the naked singularity and at spatial infinity, and the leading terms of the solutions are presented. We investigate the linear stability of the static solutions with respect to radial perturbations and numerically estimate the range of parameters corresponding to stable/unstable configurations.